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Points of Contact of Tangents from (2, 3) to y^2 = 4x | JEE Advanced Parabola

JEE Maths question with a full step-by-step solution.

Question

Q

and

R

are the points at which the tangents drawn from the point

P(2,3)

touch the parabola

y^{2} = 4x

. Find these two points.

(9,6)

and

(1,2)

(4,4)

and

(9,6)

(9,6)

and

\left(\dfrac14,\,1\right)

(1,2)

and

(4,4)

correct

Solution

Step 1: For

y^{2}=4ax

the chord of contact of the tangents from

(x_1,y_1)

T = 0

, i.e.

yy_1 = 2a(x+x_1)

. Here

a = 1

and

(x_1,y_1) = (2,3)

3y = 2(x+2), \qquad \text{i.e.} \qquad 3y = 2x + 4

Step 2: The points of contact lie on this chord and on the parabola. Putting

x = \dfrac{3y-4}{2}

y^{2}=4x

y^{2} = 4\cdot\frac{3y-4}{2} = 2(3y-4) = 6y-8

y^{2}-6y+8 = 0 \ \Longrightarrow\ (y-2)(y-4)=0 \ \Longrightarrow\ y = 2 \text{ or } y = 4

Step 3: From

x = y^{2}/4

y=2 \Rightarrow x = 1, \qquad y=4 \Rightarrow x = 4

Step 4: The points of contact are

(1,2)

and

(4,4)

. Check:

2^{2}=4(1)

and

4^{2}=4(4)

, and both satisfy

3y = 2x+4

. Answer: (4).

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